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arXiv · 1205.5165

An Overdetermined Problem in Potential Theory

Dmitry Khavinson·Erik Lundberg·Razvan Teodorescu
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Abstract

We investigate a problem posed by L. Hauswirth, F. H\'elein, and F. Pacard, namely, to characterize all the domains in the plane that admit a "roof function", i.e., a positive harmonic function which solves simultaneously a Dirichlet problem with null boundary data, and a Neumann problem with constant boundary data. Under some a priori assumptions, we show that the only three examples are the exterior of a disk, a halfplane, and a nontrivial example. We show that in four dimensions the nontrivial simply connected example does not have any axially symmetric analog containing its own axis of symmetry.

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Dmitry Khavinson, Erik Lundberg, Razvan Teodorescu. 2012-05-23. An Overdetermined Problem in Potential Theory. https://doi.org/10.2140/pjm.2013.265.85

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